U.S. patent application number 13/797554 was filed with the patent office on 2013-08-15 for computer-implemented system and method for bounding accuracy on a forecast of photovoltaic fleet power generation.
This patent application is currently assigned to CLEAN POWER RESEARCH, L.L.C.. The applicant listed for this patent is CLEAN POWER RESEARCH, L.L.C.. Invention is credited to Thomas E. Hoff.
Application Number | 20130211722 13/797554 |
Document ID | / |
Family ID | 44912505 |
Filed Date | 2013-08-15 |
United States Patent
Application |
20130211722 |
Kind Code |
A1 |
Hoff; Thomas E. |
August 15, 2013 |
Computer-Implemented System And Method For Bounding Accuracy On A
Forecast Of Photovoltaic Fleet Power Generation
Abstract
A computer-implemented system and method for bounding accuracy
on a forecast of photovoltaic fleet power generation is provided.
Measured irradiance observations for a plurality of locations are
retrieved. The measured observations include a time series recorded
at successive time periods. Forecast irradiance observations are
retrieved. Error between the forecast and the measured observations
is identified. A mean and standard deviation of the error is
determined and combined into a fleet mean and fleet standard
deviation. Sky clearness indexes are generated as a ratio of each
measured observation and clear sky irradiance. A time series of the
sky clearness indexes is formed. Fleet irradiance statistics are
determined through statistical evaluation of the sky clearness
indexes time series. A time series of power statistics is generated
as a function of the fleet irradiance statistics and photovoltaic
fleet power rating. A statistical confidence is associated with
each power statistic in the time series.
Inventors: |
Hoff; Thomas E.; (Napa,
CA) |
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Applicant: |
Name |
City |
State |
Country |
Type |
CLEAN POWER RESEARCH, L.L.C.; |
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US |
|
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Assignee: |
CLEAN POWER RESEARCH,
L.L.C.
Napa
CA
|
Family ID: |
44912505 |
Appl. No.: |
13/797554 |
Filed: |
March 12, 2013 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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13462505 |
May 2, 2012 |
8437959 |
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13797554 |
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13453956 |
Apr 23, 2012 |
8335649 |
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13462505 |
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13190442 |
Jul 25, 2011 |
8165812 |
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13453956 |
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Current U.S.
Class: |
702/3 |
Current CPC
Class: |
H02J 2203/20 20200101;
H02J 3/383 20130101; H02J 2300/24 20200101; Y04S 40/20 20130101;
Y02E 10/56 20130101; Y02E 60/00 20130101; G01W 1/10 20130101; G06Q
50/04 20130101; H02J 3/004 20200101; Y02B 10/10 20130101; G06F
17/18 20130101; H02J 3/381 20130101; Y04S 10/50 20130101; Y02A
90/10 20180101; G06Q 10/04 20130101; G01W 1/12 20130101; Y02P 90/30
20151101; G06N 7/005 20130101 |
Class at
Publication: |
702/3 |
International
Class: |
G01W 1/12 20060101
G01W001/12; G06F 17/18 20060101 G06F017/18 |
Goverment Interests
[0002] This invention was made with State of California support
under Agreement Number 722. The California Public Utilities
Commission of the State of California has certain rights to this
invention.
Claims
1. A computer-implemented system for bounding accuracy on a
forecast of photovoltaic fleet power generation, comprising the
steps of: a data storage comprising computer-readable data,
comprising: sets of measured irradiance observations for a
plurality of locations representative of a geographic region within
which a photovoltaic fleet is located, each set of the measured
irradiance observations comprising a time series recorded at
successive time periods spaced at input time intervals; and sets of
forecast irradiance observations for the plurality of locations
over the successive time periods, each forecast irradiance
observation matching one of the measured irradiance observations
for the same location; a computer comprising a processor and memory
within which code for execution by the processor is stored,
comprising: a forecast statistics module configured to identify
error between the sets of forecast irradiance observations and the
sets of measured irradiance observations for each of the locations,
and to determine a mean and standard deviation of the error for
each location and combining the means and standard deviations for
all of the locations into a fleet mean and fleet standard deviation
of the forecast error for all locations; a sky clearness module
configured to generate a set of sky clearness indexes as a ratio of
each measured irradiance observation and clear sky irradiance, and
to form a time series of the set of the sky clearness indexes for
all of the locations within the geographic region; a fleet
statistics module configured to determine fleet irradiance
statistics for the photovoltaic fleet through statistical
evaluation of the time series of the set of the sky clearness
indexes, and to generate a time series of power statistics for the
photovoltaic fleet as a function of the fleet irradiance statistics
and photovoltaic fleet power rating; and a bounding module
configured to associate a statistical confidence with each power
statistic in the time series derived from the fleet mean and fleet
standard deviation.
2. A system according to claim 1, further comprising: a variance
module configured to determine a clearness variance from the set of
the clearness indexes for the geographic region; and an adjustment
module configured to adjust the clearness variance relative to
cloud speed over the geographic region, the input time intervals,
and a physical area of each photovoltaic station in the
photovoltaic fleet, and include the clearness variance in the
statistical confidence associated with each power statistic in the
time series.
3. A system according to claim 2, further comprising: a time lag
correlation coefficient module configured to select a time
increment relating to a time resolution of the power statistics for
the photovoltaic fleet, and to express a time lag correlation
coefficient as a relationship between the power statistics for the
photovoltaic fleet starting at a beginning of a time period and the
power statistics for the photovoltaic fleet starting at the
beginning of the time period plus the time increment; and an output
time series generation module configured to apply the time lag
correlation coefficient to the power statistics for the
photovoltaic fleet over successive time increments to form a time
series of the power statistics for the photovoltaic fleet.
4. A system according to claim 3, wherein the time lag correlation
coefficient is expressed in terms of a configuration of the
photovoltaic fleet, the clearness variance, and the variance of the
change in the clearness variance associated with the input time
intervals.
5. A system according to claim 1, further comprising: an error
evaluation module configured to evaluate a relative mean absolute
error as the error between the sets of forecast irradiance
observations and the sets of measured irradiance observations for
each of the locations; a variance module configured to determine
the variance from the relative mean absolute error of each location
and combining the variances for all of the locations into a fleet
variance for all locations; and an adjustment module configured to
include the fleet variance in the statistical confidence associated
with each power statistic in the time series.
6. A system according to claim 1, further comprising: the data
storage further comprising computer-readable data, comprising:
direct irradiance observations measured from a plurality of
ground-based weather stations that are each situated proximal to
one of the locations within the geographic region; and a point
statistics module configured to assemble the direct irradiance
observations into point statistics in the sets of measured
irradiance observations for the corresponding location, each point
statistic comprising an average of all values of the direct
irradiance observations.
7. A system according to claim 1, further comprising: the data
storage further comprising computer-readable data, comprising: a
time series of power statistics from a plurality of photovoltaic
stations comprised in the photovoltaic fleet, which are each
situated proximal to one of the locations within the geographic
region; and a performance model for each of the existing
photovoltaic stations and inferring apparent irradiance as area
statistics based on the performance model selected and the time
series of power statistics; and an average point statistics module
configured to determine the sets of measured irradiance
observations as average point statistics, each average point
statistic comprising an average of all values of the apparent
irradiance.
8. A system according to claim 1, further comprising: the data
storage further comprising computer-readable data, comprising: area
solar irradiance statistics, each comprising a set of pixels from
satellite imagery for a physical area within the geographic region;
and an average point statistics module configured to convert the
area solar irradiance statistics to irradiance statistics for an
average point within the set of pixels, and to determine the sets
of measured irradiance observations as average point statistics,
each comprising an average of all values of the set of pixels.
9. A system according to claim 8, further comprising: an area
function evaluation module configured to evaluate an area function
for each pixel by solving a discrete correlation coefficient matrix
comprising correlation coefficients between point clearness indexes
selected for pairs of the points in a satellite pixel, and to set
the sets of measured irradiance observations as the solution to the
discrete correlation coefficient matrix.
10. A system according to claim 8, further comprising: an area
function evaluation module configured to evaluate an area function
for each pixel by solving probability density function based on a
distance for pairs of the points in a satellite pixel comprising
solving an integral of the probability density function for the
distance as a multiple of a correlation coefficient function at the
distance, and to set the sets of measured irradiance observations
as the solution to the probability density function.
11. A computer-implemented method for bounding accuracy on a
forecast of photovoltaic fleet power generation, comprising the
steps of: retrieving sets of measured irradiance observations for a
plurality of locations representative of a geographic region within
which a photovoltaic fleet is located, each set of the measured
irradiance observations comprising a time series recorded at
successive time periods spaced at input time intervals; retrieving
sets of forecast irradiance observations for the plurality of
locations over the successive time periods, each forecast
irradiance observation matching one of the measured irradiance
observations for the same location; identifying error between the
sets of forecast irradiance observations and the sets of measured
irradiance observations for each of the locations; determining a
mean and standard deviation of the error for each location and
combining the means and standard deviations for all of the
locations into a fleet mean and fleet standard deviation of the
forecast error for all locations; generating a set of sky clearness
indexes as a ratio of each measured irradiance observation and
clear sky irradiance; forming a time series of the set of the sky
clearness indexes for all of the locations within the geographic
region; determining fleet irradiance statistics for the
photovoltaic fleet through statistical evaluation of the time
series of the set of the sky clearness indexes; and generating a
time series of power statistics for the photovoltaic fleet as a
function of the fleet irradiance statistics and photovoltaic fleet
power rating; and associating a statistical confidence with each
power statistic in the time series derived from the fleet mean and
fleet standard deviation, wherein the steps are performed on a
suitably-programmed computer.
12. A method according to claim 11, further comprising the steps
of: determining a clearness variance from the set of the clearness
indexes for the geographic region; adjusting the clearness variance
relative to cloud speed over the geographic region, the input time
intervals, and a physical area of each photovoltaic station in the
photovoltaic fleet; and including the clearness variance in the
statistical confidence associated with each power statistic in the
time series.
13. A method according to claim 12, further comprising the steps
of: selecting a time increment relating to a time resolution of the
power statistics for the photovoltaic fleet; expressing a time lag
correlation coefficient as a relationship between the power
statistics for the photovoltaic fleet starting at a beginning of a
time period and the power statistics for the photovoltaic fleet
starting at the beginning of the time period plus the time
increment; and applying the time lag correlation coefficient to the
power statistics for the photovoltaic fleet over successive time
increments to form a time series of the power statistics for the
photovoltaic fleet.
14. A method according to claim 13, further comprising the steps
of: expressing the time lag correlation coefficient in terms of a
configuration of the photovoltaic fleet, the clearness variance,
and the variance of the change in the clearness variance associated
with the input time intervals.
15. A method according to claim 11, further comprising the steps
of: evaluating a relative mean absolute error as the error between
the sets of forecast irradiance observations and the sets of
measured irradiance observations for each of the locations;
determining the variance from the relative mean absolute error of
each location and combining the variances for all of the locations
into a fleet variance for all locations; and including the fleet
variance in the statistical confidence associated with each power
statistic in the time series.
16. A method according to claim 11, further comprising the steps
of: collecting direct irradiance observations measured from a
plurality of ground-based weather stations that are each situated
proximal to one of the locations within the geographic region; and
assembling the direct irradiance observations into point statistics
in the sets of measured irradiance observations for the
corresponding location, each point statistic comprising an average
of all values of the direct irradiance observations.
17. A method according to claim 11, further comprising the steps
of: collecting a time series of power statistics from a plurality
of photovoltaic stations comprised in the photovoltaic fleet, which
are each situated proximal to one of the locations within the
geographic region; selecting a performance model for each of the
existing photovoltaic stations and inferring apparent irradiance as
area statistics based on the performance model selected and the
time series of power statistics; and determining the sets of
measured irradiance observations as average point statistics, each
average point statistic comprising an average of all values of the
apparent irradiance.
18. A method according to claim 11, further comprising the steps
of: collecting area solar irradiance statistics, each comprising a
set of pixels from satellite imagery for a physical area within the
geographic region; converting the area solar irradiance statistics
to irradiance statistics for an average point within the set of
pixels; and determining the sets of measured irradiance
observations as average point statistics, each comprising an
average of all values of the set of pixels.
19. A method according to claim 18, further comprising the steps
of: evaluating an area function for each pixel by solving a
discrete correlation coefficient matrix comprising correlation
coefficients between point clearness indexes selected for pairs of
the points in a satellite pixel; and setting the sets of measured
irradiance observations as the solution to the discrete correlation
coefficient matrix.
20. A method according to claim 18, further comprising the steps
of: evaluating an area function for each pixel by solving
probability density function based on a distance for pairs of the
points in a satellite pixel comprising solving an integral of the
probability density function for the distance as a multiple of a
correlation coefficient function at the distance; and setting the
sets of measured irradiance observations as the solution to the
probability density function.
21. A non-transitory computer readable storage medium storing code
for executing on a computer system to perform the method according
to claim 11.
Description
CROSS-REFERENCE TO RELATED APPLICATION
[0001] This patent application is a continuation of U.S. patent
application Ser. No. 13/462,505, filed May 2, 2012, pending; which
is a continuation of U.S. Pat. No. 8,335,649, issued Dec. 18, 2012;
which is a continuation of U.S. Pat. No. 8,165,812, issued Apr. 24,
2012, the priority dates of which are claimed and the disclosures
of which are incorporated by reference.
FIELD
[0003] This application relates in general to photovoltaic power
generation fleet planning and operation and, in particular, to a
computer-implemented system and method for bounding accuracy on a
forecast of photovoltaic fleet power generation.
BACKGROUND
[0004] The manufacture and usage of photovoltaic systems has
advanced significantly in recent years due to a continually growing
demand for renewable energy resources. The cost per watt of
electricity generated by photovoltaic systems has decreased
dramatically, especially when combined with government incentives
offered to encourage photovoltaic power generation. Photovoltaic
systems are widely applicable as standalone off-grid power systems,
sources of supplemental electricity, such as for use in a building
or house, and as power grid-connected systems. Typically, when
integrated into a power grid, photovoltaic systems are collectively
operated as a fleet, although the individual systems in the fleet
may be deployed at different physical locations within a geographic
region.
[0005] Grid connection of photovoltaic power generation fleets is a
fairly recent development. In the United States, the Energy Policy
Act of 1992 deregulated power utilities and mandated the opening of
access to power grids to outsiders, including independent power
providers, electricity retailers, integrated energy companies, and
Independent System Operators (ISOs) and Regional Transmission
Organizations (RTOs). A power grid is an electricity generation,
transmission, and distribution infrastructure that delivers
electricity from supplies to consumers. As electricity is consumed
almost immediately upon production, power generation and
consumption must be balanced across the entire power grid. A large
power failure in one part of the grid could cause electrical
current to reroute from remaining power generators over
transmission lines of insufficient capacity, which creates the
possibility of cascading failures and widespread power outages.
[0006] As a result, both planners and operators of power grids need
to be able to accurately gauge on-going power generation and
consumption, and photovoltaic fleets participating as part of a
power grid are expected to exhibit predictable power generation
behaviors. Power production data is needed at all levels of a power
grid to which a photovoltaic fleet is connected, especially in
smart grid integration, as well as by operators of distribution
channels, power utilities, ISOs, and RTOs. Photovoltaic fleet power
production data is particularly crucial where a fleet makes a
significant contribution to the grid's overall energy mix.
[0007] A grid-connected photovoltaic fleet could be dispersed over
a neighborhood, utility region, or several states and its
constituent photovoltaic systems could be concentrated together or
spread out. Regardless, the aggregate grid power contribution of a
photovoltaic fleet is determined as a function of the individual
power contributions of its constituent photovoltaic systems, which
in turn, may have different system configurations and power
capacities. The system configurations may vary based on operational
features, such as size and number of photovoltaic arrays, the use
of fixed or tracking arrays, whether the arrays are tilted at
different angles of elevation or are oriented along differing
azimuthal angles, and the degree to which each system is covered by
shade due to clouds.
[0008] Photovoltaic system power output is particularly sensitive
to shading due to cloud cover, and a photovoltaic array with only a
small portion covered in shade can suffer a dramatic decrease in
power output. For a single photovoltaic system, power capacity is
measured by the maximum power output determined under standard test
conditions and is expressed in units of Watt peak (Wp). However, at
any given time, the actual power could vary from the rated system
power capacity depending upon geographic location, time of day,
weather conditions, and other factors. Moreover, photovoltaic
fleets with individual systems scattered over a large geographical
area are subject to different location-specific cloud conditions
with a consequential affect on aggregate power output.
[0009] Consequently, photovoltaic fleets operating under cloudy
conditions can exhibit variable and unpredictable performance.
Conventionally, fleet variability is determined by collecting and
feeding direct power measurements from individual photovoltaic
systems or equivalent indirectly derived power measurements into a
centralized control computer or similar arrangement. To be of
optimal usefulness, the direct power measurement data must be
collected in near real time at fine grained time intervals to
enable a high resolution time series of power output to be created.
However, the practicality of such an approach diminishes as the
number of systems, variations in system configurations, and
geographic dispersion of the photovoltaic fleet grow. Moreover, the
costs and feasibility of providing remote power measurement data
can make high speed data collection and analysis insurmountable due
to the bandwidth needed to transmit and the storage space needed to
contain collected measurements, and the processing resources needed
to scale quantitative power measurement analysis upwards as the
fleet size grows.
[0010] For instance, one direct approach to obtaining high speed
time series power production data from a fleet of existing
photovoltaic systems is to install physical meters on every
photovoltaic system, record the electrical power output at a
desired time interval, such as every 10 seconds, and sum the
recorded output across all photovoltaic systems in the fleet at
each time interval. The totalized power data from the photovoltaic
fleet could then be used to calculate the time-averaged fleet
power, variance of fleet power, and similar values for the rate of
change of fleet power. An equivalent direct approach to obtaining
high speed time series power production data for a future
photovoltaic fleet or an existing photovoltaic fleet with
incomplete metering and telemetry is to collect solar irradiance
data from a dense network of weather monitoring stations covering
all anticipated locations of interest at the desired time interval,
use a photovoltaic performance model to simulate the high speed
time series output data for each photovoltaic system individually,
and then sum the results at each time interval.
[0011] With either direct approach, several difficulties arise.
First, in terms of physical plant, calibrating, installing,
operating, and maintaining meters and weather stations is expensive
and detracts from cost savings otherwise afforded through a
renewable energy source. Similarly, collecting, validating,
transmitting, and storing high speed data for every photovoltaic
system or location requires collateral data communications and
processing infrastructure, again at possibly significant expense.
Moreover, data loss occurs whenever instrumentation or data
communications do not operate reliably.
[0012] Second, in terms of inherent limitations, both direct
approaches only work for times, locations, and photovoltaic system
configurations when and where meters are pre-installed; thus, high
speed time series power production data is unavailable for all
other locations, time periods, and photovoltaic system
configurations. Both direct approaches also cannot be used to
directly forecast future photovoltaic system performance since
meters must be physically present at the time and location of
interest. Fundamentally, data also must be recorded at the time
resolution that corresponds to the desired output time resolution.
While low time-resolution results can be calculated from high
resolution data, the opposite calculation is not possible. For
example, photovoltaic fleet behavior with a 10-second resolution
can not be determined from data collected by existing utility
meters that collect the data with a 15-minute resolution.
[0013] The few solar data networks that exist in the United States,
such as the ARM network, described in G. M. Stokes et al., "The
atmospheric radiation measurement (ARM) program: programmatic
background and design of the cloud and radiation test bed,"
Bulletin of Am. Meteorological Society 75, 1201-1221 (1994), the
disclosure of which is incorporated by reference, and the SURFRAD
network, do not have high density networks (the closest pair of
stations in the ARM network is 50 km apart) nor have they been
collecting data at a fast rate (the fastest rate is 20 seconds at
ARM network and one minute at SURFRAD network).
[0014] The limitations of the direct measurement approaches have
prompted researchers to evaluate other alternatives. Researchers
have installed dense networks of solar monitoring devices in a few
limited locations, such as described in S. Kuszamaul et al., "Lanai
High-Density Irradiance Sensor Network for Characterizing Solar
Resource Variability of MW-Scale PV System." 35.sup.th Photovoltaic
Specialists Conf, Honolulu, Hi. (Jun. 20-25, 2010), and R. George,
"Estimating Ramp Rates for Large PV Systems Using a Dense Array of
Measured Solar Radiation Data," Am. Solar Energy Society Annual
Conf. Procs., Raleigh, N.C. (May 18, 2011), the disclosures of
which are incorporated by reference. As data are being collected,
the researchers examine the data to determine if there are
underlying models that can translate results from these devices to
photovoltaic fleet production at a much broader area, yet fail to
provide translation of the data. In addition, half-hour or hourly
satellite irradiance data for specific locations and time periods
of interest have been combined with randomly selected high speed
data from a limited number of ground-based weather stations, such
as described in CAISO 2011. "Summary of Preliminary Results of 33%
Renewable Integration Study--2010," Cal. Public Util. Comm. LTPP
Docket No. R. 10-05-006 (Apr. 29, 2011) and J. Stein, "Simulation
of 1-Minute Power Output from Utility-Scale Photovoltaic Generation
Systems," Am. Solar Energy Society Annual Conf. Procs., Raleigh,
N.C. (May 18, 2011), the disclosures of which are incorporated by
reference. This approach, however, does not produce time
synchronized photovoltaic fleet variability for any particular time
period because the locations of the ground-based weather stations
differ from the actual locations of the fleet. While such results
may be useful as input data to photovoltaic simulation models for
purpose of performing high penetration photovoltaic studies, they
are not designed to produce data that could be used in grid
operational tools.
[0015] Therefore, a need remains for an approach to efficiently
estimating power output of a photovoltaic fleet in the absence of
high speed time series power production data.
SUMMARY
[0016] An approach to generating high-speed time series
photovoltaic fleet performance data is described.
[0017] One embodiment provides a computer-implemented system and
method for bounding accuracy on a forecast of photovoltaic fleet
power generation. Sets of measured irradiance observations for a
plurality of locations are retrieved. The observations are
representative of a geographic region within which a photovoltaic
fleet is located. Each set of the measured irradiance observations
includes a time series recorded at successive time periods spaced
at input time intervals. Sets of forecast irradiance observations
for the plurality of locations over the successive time periods are
also retrieved. Each forecast irradiance observation matches one of
the measured irradiance observations for the same location. Error
between the sets of forecast irradiance observations and the sets
of measured irradiance observations is identified for each of the
locations. A mean and standard deviation of the error is determined
for each location and the means and standard deviations for all of
the locations are combined into a fleet mean and fleet standard
deviation of the forecast error for all locations. A set of sky
clearness indexes are generated as a ratio of each measured
irradiance observation and clear sky irradiance. A time series of
the set of the sky clearness indexes for all of the locations
within the geographic region is formed. Fleet irradiance statistics
for the photovoltaic fleet are determined through statistical
evaluation of the time series of the set of the sky clearness
indexes. A time series of power statistics for the photovoltaic
fleet is generated as a function of the fleet irradiance statistics
and photovoltaic fleet power rating. A statistical confidence is
associated with each power statistic in the time series derived
from the fleet mean and fleet standard deviation.
[0018] Some of the notable elements of this methodology
non-exclusively include:
[0019] (1) Employing a fully derived statistical approach to
generating high-speed photovoltaic fleet production data;
[0020] (2) Using a small sample of input data sources as diverse as
ground-based weather stations, existing photovoltaic systems, or
solar data calculated from satellite images;
[0021] (3) Producing results that are usable for any photovoltaic
fleet configuration;
[0022] (4) Supporting any time resolution, even those time
resolutions faster than the input data collection rate; and
[0023] (5) Providing results in a form that is useful and usable by
electric power grid planning and operation tools.
[0024] Still other embodiments will become readily apparent to
those skilled in the art from the following detailed description,
wherein are described embodiments by way of illustrating the best
mode contemplated. As will be realized, other and different
embodiments are possible and the embodiments' several details are
capable of modifications in various obvious respects, all without
departing from their spirit and the scope. Accordingly, the
drawings and detailed description are to be regarded as
illustrative in nature and not as restrictive.
BRIEF DESCRIPTION OF THE DRAWINGS
[0025] FIG. 1 is a flow diagram showing a computer-implemented
method for bounding accuracy on a forecast of photovoltaic fleet
power generation in accordance with one embodiment.
[0026] FIG. 2 is a block diagram showing a computer-implemented
system for bounding accuracy on a forecast of photovoltaic fleet
power generation in accordance with one embodiment.
[0027] FIG. 3 is a graph depicting, by way of example, ten hours of
time series irradiance data collected from a ground-based weather
station with 10-second resolution.
[0028] FIG. 4 is a graph depicting, by way of example, the
clearness index that corresponds to the irradiance data presented
in FIG. 3.
[0029] FIG. 5 is a graph depicting, by way of example, the change
in clearness index that corresponds to the clearness index
presented in FIG. 4.
[0030] FIG. 6 is a graph depicting, by way of example, the
irradiance statistics that correspond to the clearness index in
FIG. 4 and the change in clearness index in FIG. 5.
[0031] FIGS. 7A-7B are photographs showing, by way of example, the
locations of the Cordelia Junction and Napa high density weather
monitoring stations.
[0032] FIGS. 8A-8B are graphs depicting, by way of example, the
adjustment factors plotted for time intervals from 10 seconds to
300 seconds.
[0033] FIGS. 9A-9F are graphs depicting, by way of example, the
measured and predicted weighted average correlation coefficients
for each pair of locations versus distance.
[0034] FIGS. 10A-10F are graphs depicting, by way of example, the
same information as depicted in FIGS. 9A-9F versus temporal
distance.
[0035] FIGS. 11A-11F are graphs depicting, by way of example, the
predicted versus the measured variances of clearness indexes using
different reference time intervals.
[0036] FIGS. 12A-12F are graphs depicting, by way of example, the
predicted versus the measured variances of change in clearness
indexes using different reference time intervals.
[0037] FIGS. 13A-13F are graphs and a diagram depicting, by way of
example, application of the methodology described herein to the
Napa network.
[0038] FIG. 14 is a graph depicting, by way of example, an actual
probability distribution for a given distance between two pairs of
locations, as calculated for a 1,000 meter.times.1,000 meter grid
in one square meter increments.
[0039] FIG. 15 is a graph depicting, by way of example, a matching
of the resulting model to an actual distribution.
[0040] FIG. 16 is a graph depicting, by way of example, results
generated by application of Equation (65).
[0041] FIG. 17 is a graph depicting, by way of example, the
probability density function when regions are spaced by zero to
five regions.
[0042] FIG. 18 is a graph depicting, by way of example, results by
application of the model.
DETAILED DESCRIPTION
[0043] Photovoltaic cells employ semiconductors exhibiting a
photovoltaic effect to generate direct current electricity through
conversion of solar irradiance. Within each photovoltaic cell,
light photons excite electrons in the semiconductors to create a
higher state of energy, which acts as a charge carrier for electric
current. A photovoltaic system uses one or more photovoltaic panels
that are linked into an array to convert sunlight into electricity.
In turn, a collection of photovoltaic systems can be collectively
operated as a photovoltaic fleet when integrated into a power grid,
although the constituent photovoltaic systems may actually be
deployed at different physical locations within a geographic
region.
[0044] To aid with the planning and operation of photovoltaic
fleets, whether at the power grid, supplemental, or standalone
power generation levels, high resolution time series of power
output data is needed to efficiently estimate photovoltaic fleet
power production. The variability of photovoltaic fleet power
generation under cloudy conditions can be efficiently estimated,
even in the absence of high speed time series power production
data, by applying a fully derived statistical approach. FIG. 1 is a
flow diagram showing a computer-implemented method 10 for bounding
accuracy on a forecast of photovoltaic fleet power generation in
accordance with one embodiment. The method 10 can be implemented in
software and execution of the software can be performed on a
computer system, such as further described infra, as a series of
process or method modules or steps.
[0045] Preliminarily, a time series of solar irradiance data is
obtained (step 11) for a set of locations representative of the
geographic region within which the photovoltaic fleet is located or
intended to operate, as further described infra with reference to
FIG. 3. Each time series contains solar irradiance observations
electronically recorded at known input time intervals over
successive time periods. The solar irradiance observations can
include irradiance measured by a representative set of ground-based
weather stations (step 12), existing photovoltaic systems (step
13), satellite observations (step 14), or some combination thereof.
Other sources of the solar irradiance data are possible.
[0046] Next, the solar irradiance data in the time series is
converted over each of the time periods, such as at half-hour
intervals, into a set of clearness indexes, which are calculated
relative to clear sky global horizontal irradiance. The set of
clearness indexes are interpreted into as irradiance statistics
(step 15), as further described infra with reference to FIG. 4-6.
The irradiance statistics for each of the locations is combined
into fleet irradiance statistics applicable over the geographic
region of the photovoltaic fleet. A time lag correlation
coefficient for an output time interval is also determined to
enable the generation of an output time series at any time
resolution, even faster than the input data collection rate.
[0047] Finally, power statistics, including a time series of the
power statistics for the photovoltaic fleet, are generated (step
17) as a function of the fleet irradiance statistics and system
configuration, particularly the geographic distribution and power
rating of the photovoltaic systems in the fleet (step 16). The
resultant high-speed time series fleet performance data can be used
to predictably estimate power output and photovoltaic fleet
variability by fleet planners and operators, as well as other
interested parties.
[0048] The calculated irradiance statistics are combined with the
photovoltaic fleet configuration to generate the high-speed time
series photovoltaic production data. In a further embodiment, the
foregoing methodology can be may also require conversion of weather
data for a region, such as data from satellite regions, to average
point weather data. A non-optimized approach would be to calculate
a correlation coefficient matrix on-the-fly for each satellite data
point. Alternatively, a conversion factor for performing
area-to-point conversion of satellite imagery data is described in
commonly-assigned U.S. patent application, entitled
"Computer-Implemented System and Method for Efficiently Performing
Area-To-Point Conversion of Satellite Imagery for Photovoltaic
Power Generation Fleet Output Estimation," Ser. No. 13/190,449,
filed Jul. 25, 2011, pending, the disclosure of which is
incorporated by reference.
[0049] The high resolution time series of power output data is
determined in the context of a photovoltaic fleet, whether for an
operational fleet deployed in the field, by planners considering
fleet configuration and operation, or by other individuals
interested in photovoltaic fleet variability and prediction. FIG. 2
is a block diagram showing a computer-implemented system 20 for
bounding accuracy on a forecast of photovoltaic fleet power
generation in accordance with one embodiment. Time series power
output data for a photovoltaic fleet is generated using observed
field conditions relating to overhead sky clearness. Solar
irradiance 23 relative to prevailing cloudy conditions 22 in a
geographic region of interest is measured. Direct solar irradiance
measurements can be collected by ground-based weather stations 24.
Solar irradiance measurements can also be inferred by the actual
power output of existing photovoltaic systems 25. Additionally,
satellite observations 26 can be obtained for the geographic
region. Both the direct and inferred solar irradiance measurements
are considered to be sets of point values that relate to a specific
physical location, whereas satellite imagery data is considered to
be a set of area values that need to be converted into point
values, as further described infra. Still other sources of solar
irradiance measurements are possible.
[0050] The solar irradiance measurements are centrally collected by
a computer system 21 or equivalent computational device. The
computer system 21 executes the methodology described supra with
reference to FIG. 1 and as further detailed herein to generate time
series power data 30 and other analytics, which can be stored or
provided 27 to planners, operators, and other parties for use in
solar power generation 28 planning and operations. The data feeds
29a-c from the various sources of solar irradiance data need not be
high speed connections; rather, the solar irradiance measurements
can be obtained at an input data collection rate and application of
the methodology described herein provides the generation of an
output time series at any time resolution, even faster than the
input time resolution. The computer system 21 includes hardware
components conventionally found in a general purpose programmable
computing device, such as a central processing unit, memory, user
interfacing means, such as a keyboard, mouse, and display,
input/output ports, network interface, and non-volatile storage,
and execute software programs structured into routines, functions,
and modules for execution on the various systems. In addition,
other configurations of computational resources, whether provided
as a dedicated system or arranged in client-server or peer-to-peer
topologies, and including unitary or distributed processing,
communications, storage, and user interfacing, are possible.
[0051] The detailed steps performed as part of the methodology
described supra with reference to FIG. 1 will now be described.
Obtain Time Series Irradiance Data
[0052] The first step is to obtain time series irradiance data from
representative locations. This data can be obtained from
ground-based weather stations, existing photovoltaic systems, a
satellite network, or some combination sources, as well as from
other sources. The solar irradiance data is collected from several
sample locations across the geographic region that encompasses the
photovoltaic fleet.
[0053] Direct irradiance data can be obtained by collecting weather
data from ground-based monitoring systems. FIG. 3 is a graph
depicting, by way of example, ten hours of time series irradiance
data collected from a ground-based weather station with 10-second
resolution, that is, the time interval equals ten seconds. In the
graph, the blue line 32 is the measured horizontal irradiance and
the black line 31 is the calculated clear sky horizontal irradiance
for the location of the weather station.
[0054] Irradiance data can also be inferred from select
photovoltaic systems using their electrical power output
measurements. A performance model for each photovoltaic system is
first identified, and the input solar irradiance corresponding to
the power output is determined.
[0055] Finally, satellite-based irradiance data can also be used.
As satellite imagery data is pixel-based, the data for the
geographic region is provided as a set of pixels, which span across
the region and encompassing the photovoltaic fleet.
Calculate Irradiance Statistics
[0056] The time series irradiance data for each location is then
converted into time series clearness index data, which is then used
to calculate irradiance statistics, as described infra.
[0057] Clearness Index (Kt)
[0058] The clearness index (Kt) is calculated for each observation
in the data set. In the case of an irradiance data set, the
clearness index is determined by dividing the measured global
horizontal irradiance by the clear sky global horizontal
irradiance, may be obtained from any of a variety of analytical
methods. FIG. 4 is a graph depicting, by way of example, the
clearness index that corresponds to the irradiance data presented
in FIG. 3. Calculation of the clearness index as described herein
is also generally applicable to other expressions of irradiance and
cloudy conditions, including global horizontal and direct normal
irradiance.
[0059] Change in Clearness Index (.DELTA.Kt)
[0060] The change in clearness index (.DELTA.Kt) over a time
increment of .DELTA.t is the difference between the clearness index
starting at the beginning of a time increment t and the clearness
index starting at the beginning of a time increment t, plus a time
increment .DELTA.t. FIG. 5 is a graph depicting, by way of example,
the change in clearness index that corresponds to the clearness
index presented in FIG. 4.
[0061] Time Period
[0062] The time series data set is next divided into time periods,
for instance, from five to sixty minutes, over which statistical
calculations are performed. The determination of time period is
selected depending upon the end use of the power output data and
the time resolution of the input data. For example, if fleet
variability statistics are to be used to schedule regulation
reserves on a 30-minute basis, the time period could be selected as
30 minutes. The time period must be long enough to contain a
sufficient number of sample observations, as defined by the data
time interval, yet be short enough to be usable in the application
of interest. An empirical investigation may be required to
determine the optimal time period as appropriate.
[0063] Fundamental Statistics
[0064] Table 1 lists the irradiance statistics calculated from time
series data for each time period at each location in the geographic
region. Note that time period and location subscripts are not
included for each statistic for purposes of notational
simplicity.
TABLE-US-00001 TABLE 1 Statistic Variable Mean clearness index
.mu..sub.Kt Variance clearness index .sigma..sub.Kt.sup.2 Mean
clearness index change .mu..sub..DELTA.Kt Variance clearness index
change .mu..sub..DELTA.Kt.sup.2
[0065] Table 2 lists sample clearness index time series data and
associated irradiance statistics over five-minute time periods. The
data is based on time series clearness index data that has a
one-minute time interval. The analysis was performed over a
five-minute time period. Note that the clearness index at 12:06 is
only used to calculate the clearness index change and not to
calculate the irradiance statistics.
TABLE-US-00002 TABLE 2 Clearness Clearness Index Index (Kt) Change
(.DELTA.Kt) 12:00 50% 40% 12:01 90% 0% 12:02 90% -80% 12:03 10% 0%
12:04 10% 80% 12:05 90% -40% 12:06 50% Mean (.mu.) 57% 0% Variance
(.sigma..sup.2) 13% 27%
[0066] The mean clearness index change equals the first clearness
index in the succeeding time period, minus the first clearness
index in the current time period divided by the number of time
intervals in the time period. The mean clearness index change
equals zero when these two values are the same. The mean is small
when there are a sufficient number of time intervals. Furthermore,
the mean is small relative to the clearness index change variance.
To simplify the analysis, the mean clearness index change is
assumed to equal zero for all time periods.
[0067] FIG. 6 is a graph depicting, by way of example, the
irradiance statistics that correspond to the clearness index in
FIG. 4 and the change in clearness index in FIG. 5 using a
half-hour hour time period. Note that FIG. 6 presents the standard
deviations, determined as the square root of the variance, rather
than the variances, to present the standard deviations in terms
that are comparable to the mean.
Calculate Fleet Irradiance Statistics
[0068] Irradiance statistics were calculated in the previous
section for the data stream at each sample location in the
geographic region. The meaning of these statistics, however,
depends upon the data source. Irradiance statistics calculated from
a ground-based weather station data represent results for a
specific geographical location as point statistics. Irradiance
statistics calculated from satellite data represent results for a
region as area statistics. For example, if a satellite pixel
corresponds to a one square kilometer grid, then the results
represent the irradiance statistics across a physical area one
kilometer square.
[0069] Average irradiance statistics across the photovoltaic fleet
region are a critical part of the methodology described herein.
This section presents the steps to combine the statistical results
for individual locations and calculate average irradiance
statistics for the region as a whole. The steps differs depending
upon whether point statistics or area statistics are used.
[0070] Irradiance statistics derived from ground-based sources
simply need to be averaged to form the average irradiance
statistics across the photovoltaic fleet region. Irradiance
statistics from satellite sources are first converted from
irradiance statistics for an area into irradiance statistics for an
average point within the pixel. The average point statistics are
then averaged across all satellite pixels to determine the average
across the photovoltaic fleet region.
[0071] Mean Clearness Index (.mu..sub. Kt) and Mean Change in
Clearness Index (.mu..sub. .DELTA.Kt)
[0072] The mean clearness index should be averaged no matter what
input data source is used, whether ground, satellite, or
photovoltaic system originated data. If there are N locations, then
the average clearness index across the photovoltaic fleet region is
calculated as follows.
.mu. Kt _ = i = 1 N .mu. Kt i N ( 1 ) ##EQU00001##
[0073] The mean change in clearness index for any period is assumed
to be zero. As a result, the mean change in clearness index for the
region is also zero.
.mu. .DELTA. Kt _ = 0 ( 2 ) ##EQU00002##
[0074] Convert Area Variance to Point Variance
[0075] The following calculations are required if satellite data is
used as the source of irradiance data. Satellite observations
represent values averaged across the area of the pixel, rather than
single point observations. The clearness index derived from this
data (Kt.sup.Area) may therefore be considered an average of many
individual point measurements.
Kt Area = i = 1 N Kt i N ( 3 ) ##EQU00003##
[0076] As a result, the variance of the clearness index based on
satellite data can be expressed as the variance of the average
clearness index across all locations within the satellite
pixel.
.sigma. Kt - Area 2 = VAR [ Kt Area ] = VAR [ i = 1 N Kt i N ] ( 4
) ##EQU00004##
[0077] The variance of a sum, however, equals the sum of the
covariance matrix.
.sigma. Kt - Area 2 = ( 1 N 2 ) i = 1 N j = 1 N COV [ Kt i , Kt j ]
( 5 ) ##EQU00005##
[0078] Let .rho..sup.Kt.sup.i.sup.,Kt.sup.j represents the
correlation coefficient between the clearness index at location i
and location j within the satellite pixel. By definition of
correlation coefficient, COV[Kt.sup.i,
Kt.sup.j]=.sigma..sub.Kt.sup.i.sigma..sub.Kt.sup.j.rho..sup.Kt.sup.i.sup.-
,Kt.sup.j. Furthermore, since the objective is to determine the
average point variance across the satellite pixel, the standard
deviation at any point within the satellite pixel can be assumed to
be the same and equals .sigma..sub.Kt, which means that
.sigma..sub.Kt.sup.i.sigma..sub.Kt.sup.j=.sigma..sub.Kt.sup.2 for
all location pairs. As a result, COV[Kt.sup.i,
Kt.sup.j]=.sigma..sub.Kt.sup.2.rho..sup.Kt.sup.i.sup.,Kt.sup.j.
Substituting this result into Equation (5) and simplify.
.sigma. Kt - Area 2 = .sigma. Kt 2 ( 1 N 2 ) i = 1 N j = 1 N .rho.
Kt i , Kt j ( 6 ) ##EQU00006##
[0079] Suppose that data was available to calculate the correlation
coefficient in Equation (6). The computational effort required to
perform a double summation for many points can be quite large and
computationally resource intensive. For example, a satellite pixel
representing a one square kilometer area contains one million
square meter increments. With one million increments, Equation (6)
would require one trillion calculations to compute.
[0080] The calculation can be simplified by conversion into a
continuous probability density function of distances between
location pairs across the pixel and the correlation coefficient for
that given distance, as further described supra. Thus, the
irradiance statistics for a specific satellite pixel, that is, an
area statistic, rather than a point statistics, can be converted
into the irradiance statistics at an average point within that
pixel by dividing by a "Area" term (A), which corresponds to the
area of the satellite pixel. Furthermore, the probability density
function and correlation coefficient functions are generally
assumed to be the same for all pixels within the fleet region,
making the value of A constant for all pixels and reducing the
computational burden further. Details as to how to calculate A are
also further described supra.
.sigma. Kt 2 = .sigma. Kt - Area 2 A Kt SatellitePixel ( 7 )
##EQU00007##
[0081] where:
A Kt Satellite Pixel = ( 1 N 2 ) i = 1 N j = 1 N .rho. i , j ( 8 )
##EQU00008##
[0082] Likewise, the change in clearness index variance across the
satellite region can also be converted to an average point estimate
using a similar conversion factor, A.sub..DELTA.Kt.sup.Area.
.sigma. .DELTA. Kt 2 = .sigma. .DELTA. Kt - Area 2 A .DELTA. Kt
SatellitePixel ( 9 ) ##EQU00009##
[0083] Variance of Clearness Index (.sigma..sub. Kt.sup.2) and
Variance of Change in Clearness Index (.sigma..sub.
.DELTA.Kt.sup.2)
[0084] At this point, the point statistics (.sigma..sub.Kt.sup.2
and .sigma..sub..DELTA.Kt.sup.2) have been determined for each of
several representative locations within the fleet region. These
values may have been obtained from either ground-based point data
or by converting satellite data from area into point statistics. If
the fleet region is small, the variances calculated at each
location i can be averaged to determine the average point variance
across the fleet region. If there are N locations, then average
variance of the clearness index across the photovoltaic fleet
region is calculated as follows.
.sigma. Kt _ 2 = i = 1 N .sigma. Kt i 2 N ( 10 ) ##EQU00010##
[0085] Likewise, the variance of the clearness index change is
calculated as follows.
.sigma. .DELTA. Kt _ 2 = i = 1 N .sigma. .DELTA. Kt i 2 N ( 11 )
##EQU00011##
Calculate Fleet Power Statistics
[0086] The next step is to calculate photovoltaic fleet power
statistics using the fleet irradiance statistics, as determined
supra, and physical photovoltaic fleet configuration data. These
fleet power statistics are derived from the irradiance statistics
and have the same time period.
[0087] The critical photovoltaic fleet performance statistics that
are of interest are the mean fleet power, the variance of the fleet
power, and the variance of the change in fleet power over the
desired time period. As in the case of irradiance statistics, the
mean change in fleet power is assumed to be zero.
[0088] Photovoltaic System Power for Single System at Time t
[0089] Photovoltaic system power output (kW) is approximately
linearly related to the AC-rating of the photovoltaic system (R in
units of kW.sub.AC) times plane-of-array irradiance. Plane-of-array
irradiance can be represented by the clearness index over the
photovoltaic system (KtPV) times the clear sky global horizontal
irradiance times an orientation factor (O), which both converts
global horizontal irradiance to plane-of-array irradiance and has
an embedded factor that converts irradiance from Watts/m.sup.2 to
kW output/kW of rating. Thus, at a specific point in time (t), the
power output for a single photovoltaic system (n) equals:
P.sub.t.sup.n=R.sup.nO.sub.t.sup.nKtPV.sub.t.sup.nI.sub.t.sup.Clear,n
(12)
[0090] The change in power equals the difference in power at two
different points in time.
.DELTA.P.sub.t,.DELTA.t.sup.n=R.sup.nO.sub.t+.DELTA.t.sup.nKtPV.sub.t+.D-
ELTA.t.sup.nI.sub.t+.DELTA.t.sup.Clear,n-R.sup.nO.sub.t.sup.nKtPV.sub.t.su-
p.nI.sub.t.sup.Clear,n (13)
[0091] The rating is constant, and over a short time interval, the
two clear sky plane-of-array irradiances are approximately the same
(O.sub.t+.DELTA.t.sup.nI.sub.t+.DELTA.t.sup.Clear,n.apprxeq.O.sub.t.sup.n-
I.sub.t.sup.Clear,n), so that the three terms can be factored out
and the change in the clearness index remains.
.DELTA.P.sub.t,.DELTA.t.sup.n.apprxeq.R.sup.nO.sub.t.sup.nI.sub.t.sup.Cl-
ear,n.DELTA.KtPV.sub.t.sup.n (14)
[0092] Time Series Photovoltaic Power for Single System
[0093] P.sup.n is a random variable that summarizes the power for a
single photovoltaic system n over a set of times for a given time
interval and set of time periods. .DELTA.P.sup.n is a random
variable that summarizes the change in power over the same set of
times.
[0094] Mean Fleet Power (.mu..sub.P)
[0095] The mean power for the fleet of photovoltaic systems over
the time period equals the expected value of the sum of the power
output from all of the photovoltaic systems in the fleet.
.mu. P = E [ n = 1 N R n O n KtPV n I Clear , n ] ( 15 )
##EQU00012##
[0096] If the time period is short and the region small, the clear
sky irradiance does not change much and can be factored out of the
expectation.
.mu. P = .mu. I Clear E [ n = 1 N R n O n KtPV n ] ( 16 )
##EQU00013##
[0097] Again, if the time period is short and the region small, the
clearness index can be averaged across the photovoltaic fleet
region and any given orientation factor can be assumed to be a
constant within the time period. The result is that:
.mu..sub.P=R.sup.Adj.Fleet.mu..sub.I.sub.Clear.mu..sub. Kt (17)
where .mu..sub.I.sub.Clear is calculated,
.mu. Kt _ ##EQU00014##
is taken from Equation (1) and:
R Adj . Fleet = n = 1 N R n O n ( 18 ) ##EQU00015##
[0098] This value can also be expressed as the average power during
clear sky conditions times the average clearness index across the
region.
.mu. P = .mu. P Clear .mu. Kt _ ( 19 ) ##EQU00016##
[0099] Variance of Fleet Power (.sigma..sub.P.sup.2)
[0100] The variance of the power from the photovoltaic fleet
equals:
.sigma. P 2 = VAR [ n = 1 N R n O n KtPV n I Clear , n ] ( 20 )
##EQU00017##
[0101] If the clear sky irradiance is the same for all systems,
which will be the case when the region is small and the time period
is short, then:
.sigma. P 2 = VAR [ I Clear n = 1 N R n O n KtPV n ] ( 21 )
##EQU00018##
[0102] The variance of a product of two independent random
variables X, Y, that is, VAR[XY]) equals
E[X].sup.2VAR[Y]+E[Y].sup.2VAR[X]+VAR[X]VAR[Y]. If the X random
variable has a large mean and small variance relative to the other
terms, then VAR[XY].apprxeq.E[X].sup.2VAR[Y]. Thus, the clear sky
irradiance can be factored out of Equation (21) and can be written
as:
.sigma. P 2 = ( .mu. I Clear ) 2 VAR [ n = 1 N R n KtPV n O n ] (
22 ) ##EQU00019##
[0103] The variance of a sum equals the sum of the covariance
matrix.
.sigma. P 2 = ( .mu. I Clear ) 2 ( i = 1 N j = 1 N COV [ R i KtPV i
O i , R j KtPV j O j ] ) ( 23 ) ##EQU00020##
[0104] In addition, over a short time period, the factor to convert
from clear sky GHI to clear sky POA does not vary much and becomes
a constant. All four variables can be factored out of the
covariance equation.
.sigma. P 2 = ( .mu. I Clear ) 2 ( i = 1 N j = 1 N ( R i O i ) ( R
j O j ) COV [ KtPV i , KtPV j ] ) ( 24 ) ##EQU00021##
[0105] For any i and j, COV[KtPV.sup.i,KtPV.sup.j]= {square root
over
(.sigma..sub.KtPV.sub.i.sup.2.sigma..sub.KtPV.sub.j.sup.2)}.rho..sup.Kt.s-
up.i.sup.,Kt.sup.j.
.sigma. P 2 = ( .mu. I Clear ) 2 ( i = 1 N j = 1 N ( R i O i ) ( R
j O j ) .sigma. KtPV i 2 .sigma. KtPV j 2 .rho. Kt i , Kt j ) ( 25
) ##EQU00022##
[0106] As discussed supra, the variance of the satellite data
required a conversion from the satellite area, that is, the area
covered by a pixel, to an average point within the satellite area.
In the same way, assuming a uniform clearness index across the
region of the photovoltaic plant, the variance of the clearness
index across a region the size of the photovoltaic plant within the
fleet also needs to be adjusted. The same approach that was used to
adjust the satellite clearness index can be used to adjust the
photovoltaic clearness index. Thus, each variance needs to be
adjusted to reflect the area that the i.sup.th photovoltaic plant
covers.
.sigma..sub.KtPV.sub.i.sup.2=A.sub.Kt.sup.i.sigma..sub. Kt.sup.2
(26)
[0107] Substituting and then factoring the clearness index variance
given the assumption that the average variance is constant across
the region yields:
.sigma..sub.P.sup.2=(R.sup.Adj.Fleet.mu..sub.I.sub.Clear).sup.2P.sup.Kt.-
sigma..sub. Kt.sup.2 (27)
[0108] where the correlation matrix equals:
P Kt = i = 1 N j = 1 N ( R i O i A Kt i ) ( R j O j A Kt j ) .rho.
Kt i , Kt j ( n = 1 N R n O n ) 2 ( 28 ) ##EQU00023##
[0109] R.sup.Adj.Fleet.mu..sub.I.sub.Clear in Equation (27) can be
written as the power produced by the photovoltaic fleet under clear
sky conditions, that is:
.sigma. P 2 = .mu. P Clear 2 P Kt .sigma. Kt _ 2 ( 29 )
##EQU00024##
[0110] If the region is large and the clearness index mean or
variances vary substantially across the region, then the
simplifications may not be able to be applied. Notwithstanding, if
the simplification is inapplicable, the systems are likely located
far enough away from each other, so as to be independent. In that
case, the correlation coefficients between plants in different
regions would be zero, so most of the terms in the summation are
also zero and an inter-regional simplification can be made. The
variance and mean then become the weighted average values based on
regional photovoltaic capacity and orientation.
[0111] Discussion
[0112] In Equation (28), the correlation matrix term embeds the
effect of intra-plant and inter-plant geographic diversification.
The area-related terms (A) inside the summations reflect the
intra-plant power smoothing that takes place in a large plant and
may be calculated using the simplified relationship, as further
discussed supra. These terms are then weighted by the effective
plant output at the time, that is, the rating adjusted for
orientation. The multiplication of these terms with the correlation
coefficients reflects the inter-plant smoothing due to the
separation of photovoltaic systems from one another.
[0113] Variance of Change in Fleet Power
(.sigma..sub..DELTA.P.sup.2)
[0114] A similar approach can be used to show that the variance of
the change in power equals:
.sigma. .DELTA. P 2 = .mu. P Clear 2 P .DELTA. Kt .sigma. .DELTA.
Kt _ 2 ( 30 ) ##EQU00025##
[0115] where:
P .DELTA. Kt = i = 1 N j = 1 N ( R i O i A .DELTA. Kt i ) ( R j O j
A .DELTA. Kt j ) .rho. .DELTA. Kt i , .DELTA. Kt j ( n = 1 N R n O
n ) 2 ( 31 ) ##EQU00026##
[0116] The determination of Equations (30) and (31) becomes
computationally intensive as the network of points becomes large.
For example, a network with 10,000 photovoltaic systems would
require the computation of a correlation coefficient matrix with
100 million calculations. The computational burden can be reduced
in two ways. First, many of the terms in the matrix are zero
because the photovoltaic systems are located too far away from each
other. Thus, the double summation portion of the calculation can be
simplified to eliminate zero values based on distance between
locations by construction of a grid of points. Second, once the
simplification has been made, rather than calculating the matrix
on-the-fly for every time period, the matrix can be calculated once
at the beginning of the analysis for a variety of cloud speed
conditions, and then the analysis would simply require a lookup of
the appropriate value.
Time Lag Correlation Coefficient
[0117] The next step is to adjust the photovoltaic fleet power
statistics from the input time interval to the desired output time
interval. For example, the time series data may have been collected
and stored every 60 seconds. The user of the results, however, may
want to have photovoltaic fleet power statistics at a 10-second
rate. This adjustment is made using the time lag correlation
coefficient.
[0118] The time lag correlation coefficient reflects the
relationship between fleet power and that same fleet power starting
one time interval (.DELTA.t) later. Specifically, the time lag
correlation coefficient is defined as follows:
.rho. P , P .DELTA. t = COV [ P , P .DELTA. t ] .sigma. P 2 .sigma.
P .DELTA. t 2 ( 32 ) ##EQU00027##
[0119] The assumption that the mean clearness index change equals
zero implies that
.sigma..sub.P.sub..DELTA.t.sup.2=.sigma..sub.P.sup.2. Given a
non-zero variance of power, this assumption can also be used to
show that
COV [ P , P .DELTA. t ] .sigma. P 2 = 1 - .sigma. .DELTA. P 2 2
.sigma. P 2 . ##EQU00028##
Therefore:
[0120] .rho. P , P .DELTA. t = 1 - .sigma. .DELTA. P 2 2 .sigma. P
2 ( 33 ) ##EQU00029##
[0121] This relationship illustrates how the time lag correlation
coefficient for the time interval associated with the data
collection rate is completely defined in terms of fleet power
statistics already calculated. A more detailed derivation is
described infra.
[0122] Equation (33) can be stated completely in terms of the
photovoltaic fleet configuration and the fleet region clearness
index statistics by substituting Equations (29) and (30).
Specifically, the time lag correlation coefficient can be stated
entirely in terms of photovoltaic fleet configuration, the variance
of the clearness index, and the variance of the change in the
clearness index associated with the time increment of the input
data.
.rho. P , P .DELTA. t = 1 - P .DELTA. Kt .sigma. .DELTA. Kt _ 2 2 P
Kt .sigma. Kt _ 2 _ ( 34 ) ##EQU00030##
Generate High-Speed Time Series Photovoltaic Fleet Power
[0123] The final step is to generate high-speed time series
photovoltaic fleet power data based on irradiance statistics,
photovoltaic fleet configuration, and the time lag correlation
coefficient. This step is to construct time series photovoltaic
fleet production from statistical measures over the desired time
period, for instance, at half-hour output intervals.
[0124] A joint probability distribution function is required for
this step. The bivariate probability density function of two unit
normal random variables (X and Y) with a correlation coefficient of
.rho. equals:
f ( x , y ) = 1 2 .pi. 1 - .rho. 2 exp [ - ( x 2 + y 2 - 2 .rho. xy
) 2 ( 1 - .rho. 2 ) ] ( 35 ) ##EQU00031##
[0125] The single variable probability density function for a unit
normal random variable X alone is
f ( x ) = 1 2 .pi. exp ( - x 2 2 ) . ##EQU00032##
In addition, a conditional distribution for y can be calculated
based on a known x by dividing the bivariate probability density
function by the single variable probability density
( i . e . , f ( y | x ) = f ( x , y ) f ( x ) ) . ##EQU00033##
Making the appropriate substitutions, the result is that the
conditional distribution of y based on a known x equals:
f ( y | x ) = 1 2 .pi. 1 - .rho. 2 exp [ - ( y - p x ) 2 2 ( 1 - p
2 ) ] ( 36 ) ##EQU00034##
[0126] Define a random variable
Z = Y - .rho. x 1 - .rho. 2 ##EQU00035##
and substitute into Equation (36). The result is that the
conditional probability of z given a known x equals:
f ( z | x ) = 1 2 .pi. exp ( - z 2 2 ) ( 37 ) ##EQU00036##
[0127] The cumulative distribution function for Z can be denoted by
.phi.(z*), where z* represents a specific value for z. The result
equals a probability (p) that ranges between 0 (when z*=-.infin.)
and 1 (when z*=.infin.). The function represents the cumulative
probability that any value of z is less than z*, as determined by a
computer program or value lookup.
p = .phi. ( z * ) = 1 2 .pi. .intg. - .infin. z * exp ( - z 2 2 ) z
.intg. ( 38 ) ##EQU00037##
[0128] Rather than selecting z*, however, a probability p falling
between 0 and 1 can be selected and the corresponding z* that
results in this probability found, which can be accomplished by
taking the inverse of the cumulative distribution function.
.phi..sup.-1(p)=z* (39)
[0129] Substituting back for z as defined above results in:
.phi. - 1 ( p ) = y - .rho. x 1 - .rho. 2 ( 40 ) ##EQU00038##
[0130] Now, let the random variables equal
X = P - .mu. P .sigma. P and Y = P .DELTA. t - .mu. P .DELTA. t
.sigma. P .DELTA. t , ##EQU00039##
with the correlation coefficient being the time lag correlation
coefficient between P and P.sup..DELTA.t (i.e., let
.rho.=.rho..sup.P,P.sup..DELTA.t). When .DELTA.t is small, then the
mean and standard deviations for P.sup..DELTA.t are approximately
equal to the mean and standard deviation for P. Thus, Y can be
restated as
Y .apprxeq. P .DELTA. t - .mu. P .sigma. P . ##EQU00040##
Add a time subscript to all of the relevant data to represent a
specific point in time and substitute x, y, and .rho. into Equation
(40).
.phi. - 1 ( p ) = ( P .DELTA. t t - .mu. P .sigma. P ) - .rho. P ,
P .DELTA. t ( P t - .mu. P .sigma. P ) 1 - .rho. P , P .DELTA. t 2
( 41 ) ##EQU00041##
[0131] The random variable P.sup..DELTA.t, however, is simply the
random variable P shifted in time by a time interval of .DELTA.t.
As a result, at any given time t,
P.sup..DELTA.t.sub.t=P.sub.t+.DELTA.t. Make this substitution into
Equation (41) and solve in terms of P.sub.t+.DELTA.t.
P t + .DELTA. t = .rho. P . P .DELTA. t P t + ( 1 - .rho. P , P
.DELTA. t ) .mu. P + .sigma. P 2 ( 1 - .rho. P , P .DELTA. t 2 )
.phi. - 1 ( p ) ( 42 ) ##EQU00042##
[0132] At any given time, photovoltaic fleet power equals
photovoltaic fleet power under clear sky conditions times the
average regional clearness index, that is,
P.sub.t=P.sub.t.sup.Clear Kt.sub.t. In addition, over a short time
period, .mu..sub.p.apprxeq.P.sub.t.sup.Clear .mu..sub. Kt and
.sigma..sub.P.sup.2.apprxeq.(P.sub.t.sup.Clear).sup.2P.sup.Kt.sigma..sub.
Kt.sup.2. Substitute these three relationships into Equation (42)
and factor out photovoltaic fleet power under clear sky conditions
((P.sub.t.sup.Clear)) as common to all three terms.
P t + .DELTA. t = P t Clear [ .rho. P , P .DELTA. t K t t + ( 1 -
.rho. P , P .DELTA. t ) .mu. Kt _ + P K t .sigma. 2 Kt ( 1 - .rho.
P , P .DELTA. t 2 ) .phi. - 1 ( p t ) ] ( 43 ) ##EQU00043##
[0133] Equation (43) provides an iterative method to generate
high-speed time series photovoltaic production data for a fleet of
photovoltaic systems. At each time step (t+.DELTA.t), the power
delivered by the fleet of photovoltaic systems (P.sub.t+.DELTA.t)
is calculated using input values from time step t. Thus, a time
series of power outputs can be created. The inputs include: [0134]
P.sub.t.sup.Clear--photovoltaic fleet power during clear sky
conditions calculated using a photovoltaic simulation program and
clear sky irradiance. [0135] Kt.sub.t--average regional clearness
index inferred based on P.sub.t calculated in time step t, that
is,
[0135] K t t = P t P t Clear . ##EQU00044## [0136] .mu..sub.
Kt--mean clearness index calculated using time series irradiance
data and Equation (1). [0137] .sigma..sub. Kt.sup.2--variance of
the clearness index calculated using time series irradiance data
and Equation (10). [0138] .rho..sup.P,P.sup..DELTA.t--fleet
configuration as reflected in the time lag correlation coefficient
calculated using Equation (34). In turn, Equation (34), relies upon
correlation coefficients from Equations (28) and (31). A method to
obtain these correlation coefficients by empirical means is
described in commonly-assigned U.S. patent application, entitled
"Computer-Implemented System and Method for Determining
Point-To-Point Correlation Of Sky Clearness for Photovoltaic Power
Generation Fleet Output Estimation," Ser. No. 13/190,435, filed
Jul. 25, 2011, pending, and U.S. patent application, entitled
"Computer-Implemented System and Method for Efficiently Performing
Area-To-Point Conversion of Satellite Imagery for Photovoltaic
Power Generation Fleet Output Estimation," Ser. No. 13/190,449,
filed Jul. 25, 2011, pending, the disclosure of which is
incorporated by reference. [0139] P.sup.Kt--fleet configuration as
reflected in the clearness index correlation coefficient matrix
calculated using Equation (28) where, again, the correlation
coefficients may be obtained using the empirical results as further
described infra. [0140] .phi..sup.-1(p.sub.t)--the inverse
cumulative normal distribution function based on a random variable
between 0 and 1.
Derivation of Empirical Models
[0141] The previous section developed the mathematical
relationships used to calculate irradiance and power statistics for
the region associated with a photovoltaic fleet. The relationships
between Equations (8), (28), (31), and (34) depend upon the ability
to obtain point-to-point correlation coefficients. This section
presents empirically-derived models that can be used to determine
the value of the coefficients for this purpose.
[0142] A mobile network of 25 weather monitoring devices was
deployed in a 400 meter by 400 meter grid in Cordelia Junction,
Calif., between Nov. 6, 2010, and Nov. 15, 2010, and in a 4,000
meter by 4,000 meter grid in Napa, Calif., between Nov. 19, 2010,
and Nov. 24, 2010. FIGS. 7A-7B are photographs showing, by way of
example, the locations of the Cordelia Junction and Napa high
density weather monitoring stations.
[0143] An analysis was performed by examining results from Napa and
Cordelia Junction using 10, 30, 60, 120 and 180 second time
intervals over each half-hour time period in the data set. The
variance of the clearness index and the variance of the change in
clearness index were calculated for each of the 25 locations for
each of the two networks. In addition, the clearness index
correlation coefficient and the change in clearness index
correlation coefficient for each of the 625 possible pairs, 300 of
which are unique, for each of the two locations were
calculated.
[0144] An empirical model is proposed as part of the methodology
described herein to estimate the correlation coefficient of the
clearness index and change in clearness index between any two
points by using as inputs the following: distance between the two
points, cloud speed, and time interval. For the analysis, distances
were measured, cloud speed was implied, and a time interval was
selected.
[0145] The empirical models infra describe correlation coefficients
between two points (i and j), making use of "temporal distance,"
defined as the physical distance (meters) between points i and j,
divided by the regional cloud speed (meters per second) and having
units of seconds. The temporal distance answers the question, "How
much time is needed to span two locations?"
[0146] Cloud speed was estimated to be six meters per second.
Results indicate that the clearness index correlation coefficient
between the two locations closely matches the estimated value as
calculated using the following empirical model:
.rho..sup.Kt.sup.i.sup.,Kt.sup.j=exp(C.sub.1.times.TemporalDistance).sup-
.ClearnessPower (44)
[0147] where TemporalDistance=Distance (meters)/CloudSpeed (meters
per second), Clearness Power=ln(C.sub.2.DELTA.t)-k, such that
5.ltoreq.k.ltoreq.15, where the expected value is k=9.3, .DELTA.t
is the desired output time interval (seconds), and
C.sub.1=10.sup.-3 seconds.sup.-1, and C.sub.2=1 seconds.sup.-1.
[0148] Results also indicate that the correlation coefficient for
the change in clearness index between two locations closely matches
the values calculated using the following empirical
relationship:
.rho..sup..DELTA.Kt.sup.i.sup.,.DELTA.Kt.sup.j=(.rho..sup.Kt.sup.i.sup.,-
Kt.sup.j).sup..DELTA.ClearnessPower (45)
[0149] where .rho..sup.Kt.sup.i.sup.,Kt.sup.j is calculated using
Equation (44) and
.DELTA. ClearnessPower = 1 + m C 2 .DELTA. t , ##EQU00045##
such that 100.ltoreq.m.ltoreq.200, where the expected value is
m=140.
[0150] Empirical results also lead to the following models that may
be used to translate the variance of clearness index and the
variance of change in clearness index from the measured time
interval (.DELTA.t ref) to the desired output time interval
(.DELTA.t).
.sigma. Kt .DELTA. t 2 = .sigma. Kt .DELTA. t ref 2 exp [ 1 - (
.DELTA. t .DELTA. t ref ) C 3 ] ( 46 ) .sigma. .DELTA. Kt .DELTA. t
2 = .sigma. .DELTA. Kt .DELTA. t ref 2 { 1 - 2 [ 1 - ( .DELTA. t
.DELTA. t ref ) C 3 ] } ( 47 ) ##EQU00046##
[0151] where C.sub.3=0.1.ltoreq.C.sub.3.ltoreq.0.2, where the
expected value is C.sub.3=0.15.
[0152] FIGS. 8A-8B are graphs depicting, by way of example, the
adjustment factors plotted for time intervals from 10 seconds to
300 seconds. For example, if the variance is calculated at a
300-second time interval and the user desires results at a
10-second time interval, the adjustment for the variance clearness
index would be 1.49
[0153] These empirical models represent a valuable means to rapidly
calculate correlation coefficients and translate time interval with
readily-available information, which avoids the use of
computation-intensive calculations and high-speed streams of data
from many point sources, as would otherwise be required.
Validation
[0154] Equations (44) and (45) were validated by calculating the
correlation coefficients for every pair of locations in the
Cordelia Junction network and the Napa network at half-hour time
periods. The correlation coefficients for each time period were
then weighted by the corresponding variance of that location and
time period to determine weighted average correlation coefficient
for each location pair. The weighting was performed as follows:
.rho. Kt i , Kt j _ = t = 1 T .sigma. Kt - i , j t 2 .rho. t Kt i ,
Kt j t = 1 T .sigma. Kt - i , j t 2 , and ##EQU00047## .rho.
.DELTA. Kt i , .DELTA. Kt j _ = t = 1 T .sigma. .DELTA. Kt - i , j
t 2 .rho. t .DELTA. Kt i , .DELTA. Kt j t = 1 T .sigma. .DELTA. Kt
- i , j t 2 . ##EQU00047.2##
[0155] FIGS. 9A-9F are graphs depicting, by way of example, the
measured and predicted weighted average correlation coefficients
for each pair of locations versus distance. FIGS. 10A-10F are
graphs depicting, by way of example, the same information as
depicted in FIGS. 9A-9F versus temporal distance, based on the
assumption that cloud speed was 6 meters per second. The upper line
and dots appearing in close proximity to the upper line present the
clearness index and the lower line and dots appearing in close
proximity to the lower line present the change in clearness index
for time intervals from 10 seconds to 5 minutes. The symbols are
the measured results and the lines are the predicted results.
[0156] Several observations can be drawn based on the information
provided by the FIGS. 9A-9F and 10A-10F. First, for a given time
interval, the correlation coefficients for both the clearness index
and the change in the clearness index follow an exponential decline
pattern versus distance (and temporal distance). Second, the
predicted results are a good representation of the measured results
for both the correlation coefficients and the variances, even
though the results are for two separate networks that vary in size
by a factor of 100. Third, the change in the clearness index
correlation coefficient converges to the clearness correlation
coefficient as the time interval increases. This convergence is
predicted based on the form of the empirical model because
.DELTA.Clearness Power approaches one as .DELTA.t becomes
large.
[0157] Equation (46) and (47) were validated by calculating the
average variance of the clearness index and the variance of the
change in the clearness index across the 25 locations in each
network for every half-hour time period. FIGS. 11A-11F are graphs
depicting, by way of example, the predicted versus the measured
variances of clearness indexes using different reference time
intervals. FIGS. 12A-12F are graphs depicting, by way of example,
the predicted versus the measured variances of change in clearness
indexes using different reference time intervals. FIGS. 11A-11F and
12A-12F suggest that the predicted results are similar to the
measured results.
Discussion
[0158] The point-to-point correlation coefficients calculated using
the empirical forms described supra refer to the locations of
specific photovoltaic power production sites. Importantly, note
that the data used to calculate these coefficients was not obtained
from time sequence measurements taken at the points themselves.
Rather, the coefficients were calculated from fleet-level data
(cloud speed), fixed fleet data (distances between points), and
user-specified data (time interval).
[0159] The empirical relationships of the foregoing types of
empirical relationships may be used to rapidly compute the
coefficients that are then used in the fundamental mathematical
relationships. The methodology does not require that these specific
empirical models be used and improved models will become available
in the future with additional data and analysis.
Example
[0160] This section provides a complete illustration of how to
apply the methodology using data from the Napa network of 25
irradiance sensors on Nov. 21, 2010. In this example, the sensors
served as proxies for an actual 1 kW photovoltaic fleet spread
evenly over the geographical region as defined by the sensors. For
comparison purposes, a direct measurement approach is used to
determine the power of this fleet and the change in power, which is
accomplished by adding up the 10-second output from each of the
sensors and normalizing the output to a 1 kW system. FIGS. 13A-13F
are graphs and a diagram depicting, by way of example, application
of the methodology described herein to the Napa network.
[0161] The predicted behavior of the hypothetical photovoltaic
fleet was separately estimated using the steps of the methodology
described supra. The irradiance data was measured using
ground-based sensors, although other sources of data could be used,
including from existing photovoltaic systems or satellite imagery.
As shown in FIG. 13A, the data was collected on a day with highly
variable clouds with one-minute global horizontal irradiance data
collected at one of the 25 locations for the Napa network and
specific 10-second measured power output represented by a blue
line. This irradiance data was then converted from global
horizontal irradiance to a clearness index. The mean clearness
index, variance of clearness index, and variance of the change in
clearness index was then calculated for every 15-minute period in
the day. These calculations were performed for each of the 25
locations in the network. Satellite-based data or a
statistically-significant subset of the ground measurement
locations could have also served in place of the ground-based
irradiance data. However, if the data had been collected from
satellite regions, an additional translation from area statistics
to average point statistics would have been required. The averaged
irradiance statistics from Equations (1), (10), and (11) are shown
in FIG. 13B, where standard deviation (.sigma.) is presented,
instead of variance (.sigma..sup.2) to plot each of these values in
the same units.
[0162] In this example, the irradiance statistics need to be
translated since the data were recorded at a time interval of 60
seconds, but the desired results are at a 10-second resolution. The
translation was performed using Equations (46) and (47) and the
result is presented in FIG. 13C.
[0163] The details of the photovoltaic fleet configuration were
then obtained. The layout of the fleet is presented in FIG. 13D.
The details include the location of the each photovoltaic system
(latitude and longitude), photovoltaic system rating ( 1/25 kW),
and system orientation (all are horizontal).
[0164] Equation (43), and its associated component equations, were
used to generate the time series data for the photovoltaic fleet
with the additional specification of the specific empirical models,
as described in Equations (44) through (47). The resulting fleet
power and change in power is presented represented by the red lines
in FIGS. 12E and 12F.
Probability Density Function
[0165] The conversion from area statistics to point statistics
relied upon two terms A.sub.Kt and A.sub..DELTA.Kt to calculate
.sigma..sub.Kt.sup.2 and .sigma..sub..DELTA.Kt.sup.2, respectively.
This section considers these terms in more detail. For simplicity,
the methodology supra applies to both Kt and .DELTA.Kt, so this
notation is dropped. Understand that the correlation coefficient
.rho..sup.i,j could refer to either the correlation coefficient for
clearness index or the correlation coefficient for the change in
clearness index, depending upon context. Thus, the problem at hand
is to evaluate the following relationship:
A = ( 1 N 2 ) i = 1 N j = 1 N .rho. i , j ( 48 ) ##EQU00048##
[0166] The computational effort required to calculate the
correlation coefficient matrix can be substantial. For example,
suppose that the one wants to evaluate variance of the sum of
points within a 1 square kilometer satellite region by breaking the
region into one million square meters (1,000 meters by 1,000
meters). The complete calculation of this matrix requires the
examination of 1 trillion (10.sup.12) location pair
combinations.
[0167] Discrete Formulation
[0168] The calculation can be simplified using the observation that
many of the terms in the correlation coefficient matrix are
identical. For example, the covariance between any of the one
million points and themselves is 1. This observation can be used to
show that, in the case of a rectangular region that has dimension
of H by W points (total of N) and the capacity is equal distributed
across all parts of the region that:
( 1 N 2 ) i = 1 N j = 1 N .rho. i , j = ( 1 N 2 ) [ i = 0 H - 1 j =
0 i 2 k [ ( H - i ) ( W - j ) ] .rho. d + i = 0 W - 1 j = 0 i 2 k [
( W - i ) ( H - j ) ] .rho. d ] ( 49 ) ##EQU00049##
[0169] where:
- 1 , when i = 0 and j = 0 ##EQU00050## k = 1 , when j = 0 or j = i
, 2 , when 0 < j < i ##EQU00050.2##
[0170] When the region is a square, a further simplification can be
made.
( 1 N 2 ) i = 1 N j = 1 N .rho. i , j = ( 1 N 2 ) [ i = 0 N - 1 j =
0 i 2 k ( N - i ) ( N - j ) .rho. d ] ( 50 ) ##EQU00051##
[0171] where:
0 , when i = 0 and j = 0 ##EQU00052## k = 2 , when j = 0 or j = i ,
and ##EQU00052.2## 3 , when 0 < j < i ##EQU00052.3## d = ( i
2 + j 2 ) ( Area N - 1 ) . ##EQU00052.4##
[0172] The benefit of Equation (50) is that there are
N - N 2 ##EQU00053##
rather than N.sup.2 unique combinations that need to be evaluated.
In the example above, rather than requiring one trillion possible
combinations, the calculation is reduced to one-half million
possible combinations.
[0173] Continuous Formulation
[0174] Even given this simplification, however, the problem is
still computationally daunting, especially if the computation needs
to be performed repeatedly in the time series. Therefore, the
problem can be restated as a continuous formulation in which case a
proposed correlation function may be used to simplify the
calculation. The only variable that changes in the correlation
coefficient between any of the location pairs is the distance
between the two locations; all other variables are the same for a
given calculation. As a result, Equation (50) can be interpreted as
the combination of two factors: the probability density function
for a given distance occurring and the correlation coefficient at
the specific distance.
[0175] Consider the probability density function. The actual
probability of a given distance between two pairs occurring was
calculated for a 1,000 meter.times.1,000 meter grid in one square
meter increments. The evaluation of one trillion location pair
combination possibilities was evaluated using Equation (48) and by
eliminating the correlation coefficient from the equation. FIG. 14
is a graph depicting, by way of example, an actual probability
distribution for a given distance between two pairs of locations,
as calculated for a 1,000 meter.times.1,000 meter grid in one
square meter increments.
[0176] The probability distribution suggests that a continuous
approach can be taken, where the goal is to find the probability
density function based on the distance, such that the integral of
the probability density function times the correlation coefficient
function equals:
A=.intg.f(D).rho.(d)dD (51)
[0177] An analysis of the shape of the curve shown in FIG. 14
suggests that the distribution can be approximated through the use
of two probability density functions. The first probability density
function is a quadratic function that is valid between 0 and
{square root over (Area)}.
f Quad = { ( 6 Area ) ( D - D 2 Area ) for 0 .ltoreq. D .ltoreq.
Area 0 for D > Area ( 52 ) ##EQU00054##
[0178] This function is a probability density function because
integrating between 0 and {square root over (Area)} equals 1 (i.e.,
P[0.ltoreq.D.ltoreq. {square root over (Area)}]=.intg..sub.0.sup.
{square root over (Area)}f.sub.QuadDd=1).
[0179] The second function is a normal distribution with a mean of
{square root over (Area)} and standard deviation of 0.1 {square
root over (Area)}.
f Norm = ( 1 0.1 * Area ) ( 1 2 .pi. ) - ( 1 2 ) ( D - Area 0.1 *
Area ) 2 ( 53 ) ##EQU00055##
Likewise, integrating across all values equals 1.
[0180] To construct the desired probability density function, take,
for instance, 94 percent of the quadratic density function plus 6
of the normal density function.
f=0.94.intg..sub.0.sup. {square root over
(Area)}f.sub.QuaddD+0.06.intg..sub.-.infin..sup.+.infin.f.sub.NormdD
(54)
[0181] FIG. 15 is a graph depicting, by way of example, a matching
of the resulting model to an actual distribution.
[0182] The result is that the correlation matrix of a square area
with uniform point distribution as N gets large can be expressed as
follows, first dropping the subscript on the variance since this
equation will work for both Kt and .DELTA.Kt.
A.apprxeq.[0.94.intg..sub.0.sup. {square root over
(Area)}f.sub.Quad.rho.(D)dD+0.06.intg..sub.-.infin..sup.+.infin.f.sub.Nor-
m.rho.(D)dD] (55)
[0183] where .rho.(D) is a function that expresses the correlation
coefficient as a function of distance (D).
Area to Point Conversion Using Exponential Correlation
Coefficient
[0184] Equation (55) simplifies the problem of calculating the
correlation coefficient and can be implemented numerically once the
correlation coefficient function is known. This section
demonstrates how a closed form solution can be provided, if the
functional form of the correlation coefficient function is
exponential.
[0185] Noting the empirical results as shown in the graph in FIGS.
9A-9F, an exponentially decaying function can be taken as a
suitable form for the correlation coefficient function. Assume that
the functional form of correlation coefficient function equals:
.rho. ( D ) = xD Area Let Quad be the solution to .intg. 0 Area f
Quad .rho. ( D ) D . ( 56 ) Quad = .intg. 0 Area f Quad .rho. ( D )
D = ( 6 Area ) .intg. 0 Area ( D - D 2 Area ) [ xD Area ] D
Integrate to solve . ( 57 ) Quad = ( 6 ) [ ( x Area D - 1 ) xD Area
- ( ( x Area ) 2 D 2 - 2 x Area D + 2 ) xD Area ] ( 58 )
##EQU00056##
[0186] Complete the result by evaluating at D equal to {square root
over (Area)} for the upper bound and 0 for the lower bound. The
result is:
Quad = ( 6 x 3 ) [ ( x - 2 ) ( x + 1 ) + 4 ] ( 59 )
##EQU00057##
[0187] Next, consider the solution to
.intg..sub.-.infin..sup.+.infin.f.sub.Norm.rho..sup.(D)dD, which
will be called Norm.
Norm = ( 1 .sigma. ) ( 1 2 .pi. ) .intg. - .infin. + .infin. - ( 1
2 ) ( D - .mu. .sigma. ) 2 xD Area D ( 60 ) ##EQU00058##
[0188] Where .mu.= {square root over (Area)} and .sigma.=0.1
{square root over (Area)}. Simplifying:
Norm = [ x Area ( .mu. + ( 1 2 x Area ) .sigma. 2 ) ] ( 1 .sigma. )
( 1 2 .pi. ) .intg. - .infin. + .infin. - ( 1 2 ) [ D - ( .mu. + x
Area .sigma. 2 ) .sigma. ] 2 D Substitute z = D - ( .mu. + x Area
.sigma. 2 ) .sigma. and .sigma. z = D . ( 61 ) Norm = [ x Area (
.mu. + ( 1 2 x Area ) .sigma. 2 ) ] ( 1 2 .pi. ) .intg. - .infin. +
.infin. - ( 1 2 ) z 2 z ( 62 ) ##EQU00059##
[0189] Integrate and solve.
Norm = x Area ( .mu. + ( 1 2 x Area ) .sigma. 2 ) ( 63 )
##EQU00060##
[0190] Substitute the mean of {square root over (Area)} and the
standard deviation of 0.1 {square root over (Area)} into Equation
(63).
Norm=e.sup.x(1+0.005x) (64)
[0191] Substitute the solutions for Quad and Norm back into
Equation (55). The result is the ratio of the area variance to the
average point variance. This ratio was referred to as A (with the
appropriate subscripts and superscripts) supra.
A = 0.94 ( 6 x 3 ) [ ( x - 2 ) ( x + 1 ) + 4 ] + 0.6 x ( 1 + 0.005
x ) ( 65 ) ##EQU00061##
Example
[0192] This section illustrates how to calculate A for the
clearness index for a satellite pixel that covers a geographical
surface area of 1 km by 1 km (total area of 1,000,000 m.sup.2),
using a 60-second time interval, and 6 meter per second cloud
speed. Equation (56) required that the correlation coefficient be
of the form
xD Area ##EQU00062##
. The empirically derived result in Equation (44) can be rearranged
and the appropriate substitutions made to show that the correlation
coefficient of the clearness index equals
exp [ ( ln .DELTA. t - 9.3 ) D 1000 CloudSpeed ] . ##EQU00063##
Multiply the exponent by
Area Area , ##EQU00064##
so that the correlation coefficient equals
exp { [ ( ln .DELTA. t - 9.3 ) Area 1000 CloudSpeed ] [ D Area ] }
. ##EQU00065##
This expression is now in the correct form to apply Equation (65),
where
x = ( ln .DELTA. t - 9.3 ) Area 1000 CloudSpeed . ##EQU00066##
Inserting the assumptions results in
x = ( ln 60 - 9.3 ) 1 , 000 , 000 1000 .times. 6 = - 0.86761 ,
##EQU00067##
which is applied to Equation (65). The result is that A equals 65
percent, that is, the variance of the clearness index of the
satellite data collected over a 1 km.sup.2 region corresponds to 65
percent of the variance when measured at a specific point. A
similar approach can be used to show that the A equals 27 percent
for the change in clearness index. FIG. 16 is a graph depicting, by
way of example, results generated by application of Equation
(65).
Time Lag Correlation Coefficient
[0193] This section presents an alternative approach to deriving
the time lag correlation coefficient. The variance of the sum of
the change in the clearness index equals:
.sigma. .DELTA. Kt 2 = VAR ( Kt .DELTA. t - Kt ) ( 66 )
##EQU00068##
where the summation is over N locations. This value and the
corresponding subscripts have been excluded for purposes of
notational simplicity.
[0194] Divide the summation into two parts and add several
constants to the equation:
.sigma. .DELTA. Kt 2 = VAR [ .sigma. Kt .DELTA. t ( Kt .DELTA. t
.sigma. Kt .DELTA. t ) - .sigma. Kt ( Kt .sigma. Kt ) ] ( 67 )
##EQU00069##
[0195] Since
.sigma..sub..SIGMA.Kt.sub..DELTA.t.apprxeq..sigma..sub..SIGMA.Kt
(or .sigma..sub..SIGMA.Kt.sub..DELTA.t=.sigma..sub..SIGMA.Kt if the
first term in Kt and the last term in Kt.sup..DELTA.t are the
same):
.sigma. .DELTA. Kt 2 = .sigma. Kt 2 VAR [ Kt .DELTA. t .sigma. Kt
.DELTA. t - Kt .sigma. Kt ] ( 68 ) ##EQU00070##
[0196] The variance term can be expanded as follows:
.sigma. .DELTA. Kt 2 = .sigma. Kt 2 { VAR [ Kt .DELTA. t ] .sigma.
Kt .DELTA. t 2 + VAR [ Kt ] .sigma. Kt 2 - 2 COV [ Kt , t .DELTA. t
] .sigma. Kt .sigma. Kt .DELTA. t } ( 69 ) ##EQU00071##
[0197] Since
COV[.SIGMA.Kt,.SIGMA.Kt.sup..DELTA.t]=.sigma..sub..SIGMA.Kt.sigma..sub..S-
IGMA.Kt.sub..DELTA.t.rho..sup..SIGMA.Kt,.SIGMA.Kt.sup..DELTA.t, the
first two terms equal one and the covariance term is replaced by
the correlation coefficient.
.sigma..sub..SIGMA..DELTA.Kt.sup.2=2.sigma..sub..SIGMA.Kt.sup.2(1-.rho..-
sup..SIGMA.Kt,.SIGMA.Kt.sup..DELTA.t) (70)
[0198] This expression rearranges to:
.rho. Kt , Kt .DELTA. t = 1 - ( 1 2 .sigma. .DELTA. Kt 2 .sigma. Kt
2 ) ( 71 ) ##EQU00072##
[0199] Assume that all photovoltaic plant ratings, orientations,
and area adjustments equal to one, calculate statistics for the
clearness alone using the equations described supra and then
substitute. The result is:
.rho. Kt , Kt .DELTA. t = 1 - P .DELTA. Kt .sigma. 2 .DELTA. Kt 2 P
Kt .sigma. 2 Kt ( 72 ) ##EQU00073##
Relationship Between Time Lag Correlation Coefficient and
Power/Change in Power Correlation Coefficient
[0200] This section derives the relationship between the time lag
correlation coefficient and the correlation between the series and
the change in the series for a single location.
.rho. P , .DELTA. P = COV [ P , .DELTA. P ] .sigma. P 2 .sigma.
.DELTA. P 2 = COV [ P , P .DELTA. t - P ] .sigma. P 2 .sigma.
.DELTA. P 2 = COV [ P , P .DELTA. t ] - .sigma. P 2 .sigma. P 2
.sigma. .DELTA. P 2 ##EQU00074## Since ##EQU00074.2## .sigma.
.DELTA. P 2 = VAR P .DELTA. t - P = .sigma. P 2 + .sigma. P .DELTA.
t 2 - 2 COV P , P .DELTA. t , and ##EQU00074.3## COV P , P .DELTA.
t = .rho. P , P .DELTA. t .sigma. P 2 .sigma. P .DELTA. t 2 , then
: ##EQU00074.4## .rho. P , .DELTA. P = .rho. P , P .DELTA. t
.sigma. P 2 .sigma. P .DELTA. t 2 - .sigma. P 2 .sigma. P 2 (
.sigma. P 2 + .sigma. P .DELTA. t 2 - 2 .rho. P , P .DELTA. t
.sigma. P 2 .sigma. P .DELTA. t 2 ) ##EQU00074.5##
[0201] Since
.sigma..sub.P.sup.2.apprxeq..sigma..sub.P.sub..DELTA.t.sup.2, this
expression can be further simplified. Then, square both expression
and solve for the time lag correlation coefficient:
.rho..sup.P,P.sup..DELTA.t=1-2(.rho..sup.P,.DELTA.P).sup.2
Correlation Coefficients Between Two Regions
[0202] Assume that the two regions are squares of the same size,
each side with N points, that is, a matrix with dimensions of
{square root over (N)} by {square root over (N)} points, where
{square root over (N)} is an integer, but are separated by one or
more regions. Thus:
i = 1 N j = 1 N ( 1 N 2 ) .rho. i , j = ( 1 N 2 ) [ i = 0 N - 1 j =
1 - N N - 1 k ( N - i ) ( N - j ) .rho. d ] ( 73 ) ##EQU00075##
[0203] where:
k = 1 , when i = 0 2 , when i > 0 , and ##EQU00076## d = ( i 2 +
( j + M N ) 2 ) ( Area N - 1 ) , and ##EQU00076.2##
[0204] such that M equals the number of regions.
[0205] FIG. 17 is a graph depicting, by way of example, the
probability density function when regions are spaced by zero to
five regions. FIG. 17 suggests that the probability density
function can be estimated using the following distribution:
f = { 1 - ( Spacing - D Area ) for Spacing - Area .ltoreq. D
.ltoreq. Spacing 1 + ( Spacing - D Area ) for Spacing .ltoreq. D
.ltoreq. Spacing + Area 0 all else ( 74 ) ##EQU00077##
[0206] This function is a probability density function because the
integration over all possible values equals zero. FIG. 18 is a
graph depicting, by way of example, results by application of this
model.
[0207] While the invention has been particularly shown and
described as referenced to the embodiments thereof, those skilled
in the art will understand that the foregoing and other changes in
form and detail may be made therein without departing from the
spirit and scope.
* * * * *